Theorem cocanfo 37964

Description: Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Assertion

Ref	Expression
cocanfo	⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐺 = 𝐻)

Proof of Theorem cocanfo

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 769	. . . . . 6 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹))
2	1	fveq1d 6844	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑦) = ((𝐻 ∘ 𝐹)‘𝑦))
3		simpl1 1193	. . . . . . 7 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐹:𝐴–onto→𝐵)
4		fof 6754	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
5	3, 4	syl 17	. . . . . 6 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐹:𝐴⟶𝐵)
6		fvco3 6941	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑦) = (𝐺‘(𝐹‘𝑦)))
7	5, 6	sylan 581	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑦) = (𝐺‘(𝐹‘𝑦)))
8		fvco3 6941	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
9	5, 8	sylan 581	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
10	2, 7, 9	3eqtr3d 2780	. . . 4 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)))
11	10	ralrimiva 3130	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → ∀𝑦 ∈ 𝐴 (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)))
12		fveq2 6842	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐺‘(𝐹‘𝑦)) = (𝐺‘𝑥))
13		fveq2 6842	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐻‘(𝐹‘𝑦)) = (𝐻‘𝑥))
14	12, 13	eqeq12d 2753	. . . . 5 ⊢ ((𝐹‘𝑦) = 𝑥 → ((𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)) ↔ (𝐺‘𝑥) = (𝐻‘𝑥)))
15	14	cbvfo 7245	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ 𝐴 (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
16	3, 15	syl 17	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → (∀𝑦 ∈ 𝐴 (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
17	11, 16	mpbid 232	. 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥))
18		eqfnfv 6985	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
19	18	3adant1 1131	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
20	19	adantr 480	. 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → (𝐺 = 𝐻 ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
21	17, 20	mpbird 257	1 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐺 = 𝐻)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∘ ccom 5636   Fn wfn 6495  ⟶wf 6496  –onto→wfo 6498  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508

This theorem is referenced by: (None)